= Solution
The variance at zero is zero, so $X_0=0$ almost surely. For $s<t$, the increment is Gaussian with mean zero and variance
$$
\mathbb E[(X_t-X_s)^2]=t+s-2s=t-s.
$$
For every $u\leq s$,
$$
\operatorname{Cov}(X_t-X_s,X_u)=u-u=0.
$$
The increment and any finite vector of past values are jointly Gaussian. Zero cross-covariances imply their independence, by the factorization of the joint Gaussian <characteristic function>. The <Monotone class theorem> extends this independence to the sigma-field generated by all past values, $\mathcal F_s^X=\sigma(X_u:u\leq s)$.
Together with the assumed continuity, these prove
$$
\boxed{X\text{ is a Brownian motion in its natural filtration}.}
$$
This is the <Gaussian-process characterization of Brownian motion>. Completion of the natural filtration preserves the independence assertion.
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